Solve Limiting Value with Squeeze Theorem: x-1

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In summary, the Squeeze Theorem is a mathematical theorem used to find the limit of a function by using two other functions that "squeeze" the original function between them. It works by stating that if two functions, g(x) and h(x), are both approaching the same limit as x approaches a certain value, and a third function, f(x), is always between them, then f(x) will also approach the same limit. This theorem can be used to solve limits that are difficult or impossible to solve using other methods, especially when the function approaches infinity or negative infinity. However, it has limitations, such as requiring two other functions that approach the same limit and being unable to find the value itself, only the limit as x approaches
  • #1
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How do I solve:
lim [(x^2-2x+1)cos(1/x^2-1)]=0
x-1
 
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  • #2
That limit is zero.

[tex] \left(x^{2}-2x+1\right)(-1)\leq \left(x^{2}-2x+1\right)\left(\cos\frac{1}{x^{2}-1}\right)\leq \left(x^{2}-2x+1\right)(+1) [/tex]

Take limit in all 3 of them and u'll have your answer.

Daniel.
 
  • #3


To solve this limit using the Squeeze Theorem, we need to find two functions that are both greater than or equal to the given function and whose limits as x approaches 1 are equal to 0.

One possible choice for the upper bound function is f(x) = x^2 - 2x + 1, since it is always greater than or equal to the given function and its limit as x approaches 1 is equal to 0.

For the lower bound function, we can use g(x) = -x^2 + 2x - 1, which is always less than or equal to the given function and also has a limit of 0 as x approaches 1.

Therefore, we have:

-g(x) ≤ x^2 - 2x + 1 ≤ f(x)

Taking the limit as x approaches 1 for all three functions, we get:

-lim g(x) = 0 ≤ lim (x^2 - 2x + 1) ≤ lim f(x) = 0

By the Squeeze Theorem, since the upper and lower bound functions have the same limit as x approaches 1, the given function must also have a limit of 0.

Therefore, the solution to the limit is 0.
 

1. What is the Squeeze Theorem?

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a mathematical theorem used to find the limit of a function using two other functions that "squeeze" the original function between them.

2. How does the Squeeze Theorem work?

The Squeeze Theorem states that if two functions, g(x) and h(x), are both approaching the same limit as x approaches a certain value, and a third function, f(x), is always between g(x) and h(x), then f(x) will also approach the same limit as x approaches that value.

3. How is the Squeeze Theorem used to solve limiting values?

In order to use the Squeeze Theorem to solve a limiting value, you must first identify two functions that are approaching the same limit as the original function. Then, show that the original function is always between these two functions. Finally, use the limit of the two known functions to determine the limit of the original function.

4. What is the main advantage of using the Squeeze Theorem to solve limiting values?

The main advantage of using the Squeeze Theorem is that it can be used to solve limits that are otherwise difficult or impossible to solve using other methods. It is particularly useful for finding the limit of a function as it approaches infinity or negative infinity.

5. Are there any limitations to using the Squeeze Theorem?

While the Squeeze Theorem can be a very useful tool in solving limiting values, it does have its limitations. It can only be used when there are two other functions that are approaching the same limit as the original function, and it can only be used to find the limit as x approaches a specific value, not the value itself.

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